Riemann-Stieltjes Integral Operators between Weighted Bergman Spaces

This note completely describes the bounded or compact Riemann-Stieltjes integral operators $T_g$ acting between the weighted Bergman space pairs $(A^p_\alpha,A^q_\beta)$ in terms of particular regularities of the holomorphic symbols $g$ on the open unit ball of $\Bbb C^n$

Towards Conformal Capacities in Euclidean Spaces

This paper addresses the so-called conformal capacities in $\mathbb R^n$, $n\ge 3$, through comparing three existing definitions (due to Betsakos, Colesanti-Cuoghi, Anderson-Vamananmurthy-Fuglede respectively) and studying their associated iso-capacitary inequalities with connection to half-diameter, mean-width, mean-curvature and ADM-mass, Hadamard type variational formula, Minkowski type problem, and Yau type problem.Comment: 33 page

Optimal Monotonicity of LpL^p Integral of Conformal Invariant Green Function

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $\Omega$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of total Gauss curvature of a conformal metric on $\Omega$. Consequently, new analytic and geometric isoperimetric-type inequalities are discovered. Furthermore, when applying the geometric principle to two-dimensional Riemannian manifolds, we find fortunately that $\{0,1\}$-form of the induced principle is midway between Moser-Trudinger's inequality and Nash-Sobolev's inequality on complete noncompact boundary-free surfaces, and yet equivalent to Nash-Sobolev's/Faber-Krahn's eigenvalue/Heat-kernel-upper-bound/Log-Sobolev's inequality on the surfaces with finite total Gauss curvature and quadratic area growth.Comment: 25 page

p-capacity vs surface-area

This paper is devoted to exploring the relationship between the $[1,n)\ni p$-capacity and the surface-area in $\mathbb R^{n\ge 2}$ which especially shows: if $\Omega\subset\mathbb R^n$ is a convex, compact, smooth set with its interior $\Omega^\circ\not=\emptyset$ and the mean curvature $H(\partial\Omega,\cdot)>0$ of its boundary $\partial\Omega$ then $$\left(\frac{n(p-1)}{p(n-1)}\right)^{p-1}\le\frac{\left(\frac{\hbox{cap}_p(\Omega)}{\big(\frac{p-1}{n-p}\big)^{1-p}\sigma_{n-1}}\right)}{\left(\frac{\hbox{area}(\partial\Omega)}{\sigma_{n-1}}\right)^\frac{n-p}{n-1}}\le\left(\sqrt[n-1]{\int_{\partial\Omega}\big(H(\partial\Omega,\cdot)\big)^{n-1}\frac{d\sigma(\cdot)}{\sigma_{n-1}}}\right)^{p-1}\quad\forall\quad p\in (1,n)$$ whose limits $1\leftarrow p\ \&\ p\rightarrow n$ imply $$1=\frac{cap_1(\Omega)}{\hbox{area}(\partial\Omega)}\ \ \& \ \int_{\partial\Omega}\big(H(\partial\Omega,\cdot)\big)^{n-1}\frac{d\sigma(\cdot)}{\sigma_{n-1}}\ge 1,$$ thereby not only discovering that the new best known constant is roughly half as far from the one conjectured by P\'olya-Szeg\"o in \cite[(2)]{P} but also extending the P\'olya-Szeg\"o inequality in \cite[(5)]{P}, with both the conjecture and the inequality being stated for the electrostatic capacity of a convex solid in $\mathbb R^3$.Comment: 13 page

The Qp\mathcal{Q}_p Carleson Measure Problem

Let $\mu$ be a nonnegative Borel measure on the open unit disk $\mathbb{D}\subset\mathbb{C}$. This note shows how to decide that the M\"obius invariant space $\mathcal{Q}_p$, covering $\mathcal{BMOA}$ and $\mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T^\infty_p(\mu)$. Interestingly, the embedding result can be used to determine the boundedness (resp., the compactness) of the Volterra-type and multiplication operators on $\mathcal{Q}_p$.Comment: 12 page

Optimal geometric estimates for fractional Sobolev capacities

This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.Comment: 6 page

A Maximum Problem of S.-T. Yau for Variational p-Capacity

Through using the semidiameter (in connection to: the mean radius and surface radius) of a convex closed hypersurface in $\mathbb R^{n\ge 2}$ as an sharp upper bound of the variational $(1,n)\ni p$-capacity radius, this paper settles a restriction/variant of S.-T. Yau's \cite[Problem 59]{Yau} from the surface area to the variational $p$-capacity whose limit as $p\to 1$ actually induces the surface area.Comment: 17 pages, accepted by Advances in Geometry, 201

Geometric Capacity Potentials on Convex Plane Rings

Under $1<p\le 2$, this paper presents some old and new convexity/isoperimetry based inequalities for the variational $p$-capacity potentials on convex plane rings.Comment: 10 page

Complex Lie algebras corresponding to weighted projective lines

The aim of this paper is to give an alternative proof of Kac's theorem for weighted projective lines (\cite{W}) over the complex field. The geometric realization of complex Lie algebras arising from derived categories (\cite{XXZ}) is essentially used.Comment: 8 page

Two Predualities and Three Operators over Analytic Campanato Spaces

This article is devoted to not only characterizing the first and second preduals of the analytic Campanato spaces ($\mathcal{CA}_p)$ on the unit disk, but also investigating boundedness of three operators: superposition ($\mathsf{S}^\phi$); backward shift ($\mathsf{S_b}$); Schwarzian derivative ($\mathsf{S}$), acting on $\mathcal{CA}_p$.Comment: 17 page